CircleHard

Question

The angle brtween a pair of tangents drawn from a point R to the circle x2 + y2 - 4x - 6y + 9sin2α + 13cos2α = 0 is 2α. The equation of the locus of the point P is

Options

A.x2 + y2 + 4x - 6y + 4 = 0
B.x2 + y2 + 4x - 6y - 9 = 0
C.x2 + y2 + 4x - 6y - 4 = 0
D.x2 + y2 + 4x - 6y + 9 = 0

Solution

       
Centre of the circle
x2 + y2 + 4x - 6y + 9 sin2α + 13cos2α = 0
is   C (-2, 3) and its radius is
       
       
       
Let (h, k) be any point P and ∠APC = α, ∠PAC = π / 2
That is, triangle APC is a right triangle.
∴    
⇒     (h + 2)2 + (k - 3)2 = 4
⇒     h2 + 4 + 4h + k2 + 9 - 6k = 4
⇒     h2 + k2 + 4h - 6k + 9 = 0
Thus, required equation of the locus is
        x2 + y2 + 4x - 6y + 9 = 0

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